
A direct calculation [1]} shows that \(\operatorname{\mathbb {E}}\Vert \Sigma ^{-1/2}X\Vert ^4 \lesssim d^2\)  under our assumption. Indeed, fixing \(\omega _i = \Sigma ^{-1/2} e_i\)  the following holds for \(i = 1, \ldots , d\) :
\(\operatorname{\mathbb {E}}\left\langle \Sigma ^{-1/2}e_i, X \right\rangle ^4 \le c\left(\operatorname{\mathbb {E}}\left\langle \Sigma ^{-1/2}e_i, X \right\rangle ^2\right)^2 = c.\) 
